Permutations and Combinations.

Permutations and Combinations. A combination, in Mathematics, is a selection of a number of objects from a given set of objects, without any regard to the order in which they are placed. The objects are called elements, and the combinations are divided into classes, according to the number of elements in each. Let the given elements be the four letters $a,\; b,\; c,\; d$; the binary combinations, or selections of two, are $ab,\; ac,\; ad,\; bc,\; bd,\; cd$—six in all; the combinations of three are $abc,\; abd,\; acd,\; bcd$—four in all; while there is only one combination of four—viz. $abcd$.

Permutation, again, has reference to the order of arrangement; thus, the two elements, $a$ and $b$, may stand $ab$ or $ba$, so that every combination of two gives two permutations; the three elements, $a,\; b$, and $c$, may stand $abc,\; acb,\; bac,\; bca,\; cab,\; cba$, one combination of three thus affording six permutations. The combinations of any order with all their permutations are called the Variations. Formulas are given in works of algebra for calculating the number of permutations or combinations in any given case. Suppose seven lottery-tickets marked 1, 2, 3, to 7, and that two are to be drawn; if it is asked how many possible pairs of numbers there are, this is a question of the number of combinations of seven elements, two together, which is found to be 21. If we want to know how many times the same seven persons could sit down to table together with a different arrangement each time, this is to ask how many permutations seven objects admit of, and the formula gives $7\; \backslash times\; 6\; \backslash times\; 5\; \backslash times\; 4\; \backslash times\; 3\; \backslash times\; 2\; =\; 5040$. The theory of probabilities is founded on the laws of combination. Thus, in the case of drawing two tickets out of seven, since there are 21 possible pairs, the chance or probability of drawing any particular pair is 1 in 21, or $\backslash frac\{1\}\{21\}$. In working out questions in 'combinations' advantage is often taken of the fact that, whatever number of elements be taken from a group to form a combination, the number left gives the same number of combinations; thus, the number of combinations of 10 elements three together, is the same as that of 10 elements seven together.